Almost and weakly almost periodic functions on the unitary groups of von Neumann algebras
aa r X i v : . [ m a t h . OA ] S e p ALMOST AND WEAKLY ALMOST PERIODICFUNCTIONS ON THE UNITARY GROUPS OF VONNEUMANN ALGEBRAS
PAUL JOLISSAINT
Abstract.
Let M Ă B p H q be a von Neumann algebra acting on the(separable) Hilbert space H . We first prove that M is finite if and onlyif, for every x P M and for all vectors ξ, η P H , the coefficient function u ÞÑ x uxu ˚ ξ | η y is weakly almost periodic on the topological group U M of unitaries in M (equipped with the weak operator topology). Themain device is the unique invariant mean on the C ˚ -algebra WAP p U M q of weakly almost periodic functions on U M . Next, we prove that everycoefficient function u ÞÑ x uxu ˚ ξ | η y is almost periodic if and only if M is a direct sum of a diffuse, abelian von Neumann algebra and finite-dimensional factors. Incidentally, we prove that if M is a diffuse vonNeumann algebra, then its unitary group is minimally almost periodic. Dedicated to Pierre de la Harpe Introduction
The present work is inspired by the article [5] where P. de la Harpe provedthat, if M is a von Neumann algebra with separable predual, then it isApproximately Finite-Dimensional (AFD) if and only if there exists a leftinvariant mean on the C ˚ -algebra C b,r p U M q of right uniformly continuousfunctions on the unitary group U M of M ; in other words, M is AFD if andonly if the Polish group U M is amenable. In fact, he used the existenceof a left invariant mean on C b,r p U M q to show that M has Schwartz’s prop-erty P [16, Definition 1], the latter being equivalent to injectivity hence toapproximate finite-dimensionality by [3].For every topological group G , there is a space of continuous functions(in fact a C ˚ -algebra) on which a unique bi-invariant mean always exists:it is the set of all weakly almost periodic functions on G . The aim of thepresent article is then to exploit the existence of such a mean on the group U M . It turns out that it provides a characterization of finite von Neumannalgebras. Date : September 3, 2020.2010
Mathematics Subject Classification.
Primary 46L10, 11K70, 22D25; Secondary22A10.
Key words and phrases.
Almost periodic functions, minimally almost periodic groups,unitary groups, finite von Neumann algebras, invariant means, conditional expectations.
In order to present the content of this article, let us recall some definitionsand fix notation.First, let G be a topological group. We denote by C b p G q the C ˚ -algebraof all bounded, continuous, complex-valued functions on G equipped withthe uniform norm } f } : “ sup s P G | f p s q| . For g P G and f : G Ñ C , wedenote by g ¨ f : G Ñ C (resp. f ¨ g ) the left (resp. right) translate of f by g , i.e. p g ¨ f qp s q “ f p g ´ s q and p f ¨ g qp s q “ f p sg q for all f : G Ñ C and g, s P G . The corresponding left (resp. right) orbitis denoted by Gf (resp. f G ). A function f P C b p G q is right uniformlycontinuous if } g ¨ f ´ f } Ñ g Ñ
1. The subset of all right uniformlycontinuous functions f P C b p G q is a C ˚ -subalgebra of C b p G q denoted by C b,r p G q , and it contains all right translates of all its elements. Our definitionfollows that of P. de la Harpe [5] and F.P. Greenleaf [11], but not that of P.Eymard [8] for instance.A function f P C b,r p G q is weakly almost periodic if its orbit Gf is weaklyrelatively compact in C b,r p G q . An application of Hahn-Banach Theoremshows that the orbit is weakly relatively compact in C b,r p G q if and only if itis weakly relatively compact in the larger Banach space C b p G q .It follows from [12, Proposition 7] that Gf is weakly relatively compactif and only if f G is (see also Proposition 5.3 in the appendix). The set ofall weakly almost periodic functions on G is denoted by WAP p G q ; it is a C ˚ -subalgebra of C b,r p G q , and its main feature, which will play a centralrole here, is the existence of a unique left and right G -invariant mean m onWAP p G q . We are indebted to S. Knudby for having indicated to us thatF.P. Greenleaf’s monograph [11] contains a proof of that result for locallycompact groups, which relies on Ryll-Nardzewski Theorem, but we thinkthat it is worth presenting (for the reader’s convenience) a self-containedproof for arbitrary topological groups in the appendix of the present article.Our proof also uses Ryll-Nardzewski Theorem.Next, let M Ă B p H q be a von Neumann algebra acting on the separableHilbert space H . Our references for von Neumann algebras are the mono-graphs [7] and [19]. We denote by U M the group of unitary elements of M .It is a topological group, and even a Polish group, when endowed with anyof the following equivalent topologies on it: the weak, ultraweak, strong andultrastrong operator topologies.For T P B p H q and ξ, η P H , we define the associated coefficient function ξ ‹ T ‹ η on U M as follows: ξ ‹ T ‹ η p u q : “ x uT u ˚ ξ | η y for every u P U M .Here is the first of our main results; its proof is contained in § Theorem 1.1.
Let M Ă B p H q be a von Neumann algebra. Then it isfinite if and only if, for every x P M , all its coefficient functions ξ ‹ x ‹ η are WEAKLY) ALMOST PERIODIC FUNCTIONS AND VON NEUMANN ALGEBRAS 3 weakly almost periodic. If it is the case, there exists a conditional expectation E M : C ˚ p M, M q Ñ M whose restriction to M coincides with the canonicalcentre-valued trace on M . The previous theorem relies on the study of the space wap H p M q of op-erators on H whose all coefficient functions are weakly almost periodic on U M . The next section, which deals with an arbitrary von Neumann algebra M Ă B p H q , is devoted to the following result and some consequences. SeeTheorem 2.4 for a more detailed version. Theorem 1.2.
The set wap H p M q is a norm-closed, unital, selfadjoint sub-space of B p H q which contains the commutant M of M and the ideal K p H q of compact operators, and it is an M -bimodule. In particular, it is spannedby its positive elements. Moreover, there exists a linear, positive, unital map E : wap H p M q Ñ M such that: (1) For all T P wap H p M q and x , y P M , one has E p x T y q “ x E p T q y . (2) For every T P wap H p M q and every v P U M , the operator vT v ˚ belongs to wap H p M q and E p vT v ˚ q “ E p T q . As will be seen in §
2, the existence of E comes from the unique bi-invariantmean m on WAP p U M q . Remark . There is a priori no reason that wap H p M q be a C ˚ -algebra.However, the case of wap H p B p H qq is completely described as the followingresult shows. Theorem 1.4.
The operator system wap H p B p H qq is equal to K p H q ` C . The penultimate section is devoted to the case of operators whose coeffi-cient functions are almost periodic functions on U M : recall that f P C b,r p G q is almost periodic if the orbits Gf and f G are relatively compact for thenorm topology. We denote by AP p G q the subset (in fact the C ˚ -subalgebra)of WAP p G q of all almost periodic functions on G . Analogously, we denoteby ap H p M q the set of all operators T P B p H q whose all coefficient func-tions are almost periodic. Then we prove the following result which relieson Theorem 1.1. Theorem 1.5.
Let M Ă B p H q be a von Neumann algebra. Then M iscontained in ap H p M q if and only if M is a direct sum A ‘ À k ě M k where A is an abelian, diffuse von Neumann algebra, and where each M k is afinite-dimensional factor. In other words, M Ă ap H p M q if and only if M is a direct sum of von Neumann algebras each of which is either abelian orfinite-dimensional. A von Neumann algebra which is a direct sum of von Neumann algebraseach of which is either abelian or finite-dimensional is a strongly finite vonNeumann algebra in the sense of [10] where the authors prove that thesealgebras are characterized by the relative compactness in the norm topology
PAUL JOLISSAINT of the orbits t ϕ ˝ α : α P Int p M qu for all normal linear functionals ϕ P M ˚ .We are grateful to P. de la Harpe for having indicated that reference.Theorem 1.5 rests on the following theorem. See Theorem 4.4 for a moreprecise statement. Theorem 1.6.
Let M be a diffuse von Neumann algebra. Then its unitarygroup U M is minimally almost periodic, i.e. the only continuous, finite-dimensional, irreducible, unitary representation of U M is the trivial repre-sentation. As explained previously, the last section is an appendix which is devotedto remind the reader of properties of weakly almost periodic functions on anarbitrary topological group G , and mostly the existence of the bi-invariantmean m on WAP p G q , which plays a central role in this article, as alreadymentioned. Acknowledgements.
We are very grateful to P. de la Harpe for his numer-ous comments, questions, suggestions and references in the last versions ofthis article. 2.
The operator system wap H p M q Let H be a separable Hilbert space and let M Ă B p H q be a von Neumannalgebra acting on H .The next lemma contains general properties of coefficient functions, andin particular formulas for the left and right translates by elements of U M .Its proof uses straighforward computations which are left to the reader. Lemma 2.1.
Let T P B p H q , ξ, η P H and v P U M . Then the followingformulas hold: (2.1) ξ ‹ T ˚ ‹ η “ η ‹ T ‹ ξ, (2.2) v ¨ p ξ ‹ T ‹ η q “ p vξ q ‹ T ‹ p vη q and (2.3) p ξ ‹ T ‹ η q ¨ v “ ξ ‹ p vT v ˚ q ‹ η. Let furthermore x , y P M . Then (2.4) ξ ‹ p x T y q ‹ η “ p y ξ q ‹ T ‹ p x η q . Our next lemma is essentially [5, Lemme 1], but we provide a proof forthe sake of completeness.
Lemma 2.2.
Let T P B p H q and ξ, η P H . Then ξ ‹ T ‹ η P C b,r p U M q .Moreover, T P M if and only if all its associated coefficient functions areconstant.Furthermore, let ϕ ξ,η : U M Ñ C be defined by ϕ ξ,η p u q “ x uξ | η y p u P U M q . Then ϕ ξ,η P C b,r p U M q . WEAKLY) ALMOST PERIODIC FUNCTIONS AND VON NEUMANN ALGEBRAS 5
Proof.
Let us set ϕ : “ ξ ‹ T ‹ η for short. Then one has, for every u P U M ,by equality (2.2): |p v ¨ ϕ qp u q ´ ϕ p u q| “ |x uT u ˚ vξ | vη y ´ x uT u ˚ ξ | η y|ď |x uT u ˚ p vξ ´ ξ q| vη y| ` |x uT u ˚ ξ | vη ´ η y|ď } T }} vξ ´ ξ }} η } ` } T }} ξ }} vη ´ η } hence } v ¨ ϕ ´ ϕ } ď } T } max p} ξ } , } η }qp} vξ ´ ξ } ` } vη ´ η }q which proves that } v ¨ ϕ ´ ϕ } Ñ v Ñ M is obvious.Finally, we have for every v P U M } v ¨ ϕ ξ,η ´ ϕ ξ,η } “ sup u P U M |x v ˚ uξ | η y ´ x uξ | η y|ď } ξ }} vη ´ η } which proves that ϕ ξ,η P C b,r p U M q . (cid:3) Definition 2.3.
A linear, bounded operator T P B p H q is almost periodicwith respect to M (resp. weakly almost periodic with respect to M ) if, forall ξ, η P H , the coefficient function ξ ‹ T ‹ η belongs to AP p U M q (resp.WAP p U M q ). The set of all almost periodic operators with respect to M is denoted by ap H p M q , and similarly the set of all weakly almost periodicoperators with respect to M is denoted by wap H p M q .We focus on wap H p M q because of the existence of the unique invari-ant mean m on WAP p U M q , which implies the existence of a positive, M -bimodular map E from wap H p M q onto M . More precisely, one has thefollowing result. Theorem 2.4.
The set wap H p M q has the following properties: (a) It is a norm-closed, unital operator system in the sense of [14, Chap-ter 2] : it is a closed, selfadjoint subspace of B p H q which contains ,thus it is spanned by its positive elements. (b) For every T P wap H p M q and for all x , y P M , one has x T y P wap H p M q . In particular, M Ă wap H p M q . (c) The ideal K p H q of all linear, compact operators on H is containedin wap H p M q .Furthermore, there exists a linear, bounded and unital map E : wap H p M q Ñ B p H q which is characterized by the equality (2.5) x E p T q ξ | η y “ m p ξ ‹ T ‹ η q p ξ, η P H q , and which possesses the following properties: (1) E is a positive map. (2) For every T P wap H p M q , one has E p T q P M . PAUL JOLISSAINT (3)
For every T P wap H p M q and all x , y P M , one has E p x T y q “ x E p T q y . (4) For every T P wap H p M q , E p T q belongs to K T , where the latter de-notes the weakly closed convex hull of the orbit t uT u ˚ : u P U M u . (5) For every T P wap H p M q and every v P U M , the operator vT v ˚ belongs to wap H p M q , and E p vT v ˚ q “ E p T q . (6) For every C ˚ -subalgebra A of wap H p M q , the restriction of E to A iscompletely positive.Proof. (a) As WAP p U M q is a C ˚ -subalgebra of C b,r p U M q , wap H p M q is anorm-closed subspace of B p H q because, if } T n ´ T } Ñ n Ñ8 } ξ ‹ T n ‹ η ´ ξ ‹ T ‹ η } ď } T n ´ T }} ξ }} η } Ñ n Ñ8 . The fact that T ˚ P wap H p M q for every T P wap H p M q follows from equation(2.1) of Lemma 2.1. Hence wap H p M q is spanned by its selfadjoint elements,and, as 1 P wap H p M q , if T “ T ˚ P wap H p M q , then the equality T “ p} T } ` T q ´ p} T } ´ T q shows that wap H p M q is spanned by its positive elements.(b) We have M Ă wap H p M q thanks to Lemma 2.2. Moreover, wap H p M q isan M -bimodule by equation (2.4) of Lemma 2.1.(c) As wap H p M q is a closed subspace of B p H q , in order to show that K p H q Ă wap H p M q , it suffices to prove that wap H p M q contains all rank one operators.Thus, let ζ, ω P H and let T ζ,ω be the rank one operator defined by T ζ,ω p ξ q : “ x ξ | ω y ζ p ξ P H q . Then we have for all ξ, η P H and u, v P U M : v ¨ p ξ ‹ T ζ,ω ‹ η qp u q “ p ξ ‹ T ζ,ω ‹ η qp v ˚ u q “ x v ˚ uT ζ,ω p u ˚ vξ q| η y“ x v ˚ u x u ˚ vξ | ω y ζ | η y “ x u ˚ vξ | ω y x v ˚ uζ | η y“ x v ˚ uω | ξ y x v ˚ uζ | η y“ v ¨ p ϕ ω,ξ ϕ ζ,η qp u q where we set ϕ ξ,η : u ÞÑ x uξ | η y for all ξ, η P H . As WAP p U M q is a ˚ -algebra, in order to show that the left orbit U M p ξ ‹ T ζ,ω ‹ η q is relativelyweakly compact, it suffices to prove that, for all ξ, η P H , the orbit U M ϕ ξ,η is relatively weakly compact. Thus, let us fix ξ, η P H . One has for all u, v P U M : p v ¨ ϕ ξ,η qp u q “ x v ˚ uξ | η y “ x uξ | vη y “ ϕ ξ,vη p u q . Since ϕ ξ,η P C b,r p U M q by Lemma 2.2 and since the orbit t vη : v P U M u is weakly relatively compact in H , it suffices to prove that the map η ÞÑ ϕ ξ,η is continuous when H and C b,r p U M q are equipped with their respectiveweak topologies. Thus, let µ be a continuous linear functional on C b,r p U M q .The sesquilinear form p ζ, ω q ÞÑ µ p ϕ ζ,ω q satisfies the following inequality: WEAKLY) ALMOST PERIODIC FUNCTIONS AND VON NEUMANN ALGEBRAS 7 | µ p ϕ ζ,ω q| ď } µ }} ζ }} ω } . Hence there exists a unique operator T µ P B p H q such that µ p ϕ ζ,ω q “ x T µ ζ | ω y for all ζ, ω P H .Now, if p η n q Ă H converges weakly to η , we have µ p ϕ ξ,η n q “ x T µ ξ | η n y Ñ n Ñ8 x T µ ξ | η y “ µ p ϕ ξ,η q . This ends the proof of the fact that all rank one operators (hence all compactoperators) belong to wap H p M q .Let us now prove the existence of the map E and all its stated properties.For T P wap H p M q , the sesquilinear form p ξ, η q ÞÑ m p ξ ‹ T ‹ η q is continuoussince one has } ξ ‹ T ‹ η } ď } T }} ξ }} η } . Hence this proves the existenceand uniqueness of E p T q for every T P wap H p M q , as well as its linearity andboundedness.(1) If T P wap H p M q is a positive operator and if ξ P H , then ξ ‹ T ‹ ξ p u q “ x uT u ˚ ξ | ξ y “ x T u ˚ ξ | u ˚ ξ y ě , which implies that E p T q ě m is a positive functional.(2) Let T P wap H p M q , v P U M and ξ, η P H . Then, using equality (2.2) andleft invariance of m , we get x v ˚ E p T q vξ | η y “ x E p T q vξ | vη y “ m pp vξ q ‹ T ‹ p vη qq“ m p v ¨ p ξ ‹ T ‹ η qq “ m p ξ ‹ T ‹ η q“ x E p T q ξ | η y , which shows that v ˚ E p T q v “ E p T q for every v P U M , thus E p T q P M .(3) follows from equality 2.4.(4) We could reproduce the proof of statement (iii) of [5, Lemme 2], butwe present a different one, based on the following property of the mean m (property (d) of Theorem 5.5): For every weakly almost periodic function f on U M , its mean m p f q belongs to the norm-closed convex hull of its rightorbit f U M . Thus, if ξ , . . . , ξ n , η , . . . , η n P H and ε ą s , . . . , s m ą ř i s i “
1, and v , . . . , v m P U M such that(2.6) ››› m ´ ÿ j ξ j ‹ T ‹ η j ¯ ´ ÿ i s i ´ ÿ j ξ j ‹ T ‹ η j ¯ ¨ v i ››› ď ε. By equality (2.3), one has ÿ i s i ´ ÿ j ξ j ‹ T ‹ η j ¯ ¨ v i “ ÿ i s i ´ ÿ j ξ j ‹ p v i T v ˚ i q ‹ η j ¯ , which yields, when evaluated at u “ ÿ i s i ´ ÿ j ξ j ‹ p v i T v ˚ i q ‹ η j ¯ p q “ ÿ i s i ´ ÿ j x v i T v ˚ i ξ j | η j y ¯ “ ÿ j x ´ ÿ i s i v i T v ˚ i ¯ ξ j | η j y PAUL JOLISSAINT As m ´ ř j ξ j ‹ T ‹ η j ¯ “ ř j x E p T q ξ j | η j y , we get ˇˇˇˇˇÿ j x E p T q ξ j | η j y ´ ÿ j x ´ ÿ i s i v i T v ˚ i ¯ ξ j | η j y ˇˇˇˇˇ ď ε which proves the claim.(5) For ξ, η P H , equality (2.3) shows that the right orbit p ξ ‹p vT v ˚ q‹ η q U M “pp ξ ‹ T ‹ η q¨ v q U M is weakly relatively compact, hence that vT v ˚ P wap H p M q .As m is right invariant, we get x E p vT v ˚ q ξ | η y “ m pp ξ ‹ T ‹ η q ¨ v q “ m p ξ ‹ T ‹ η q “ x E p T q ξ | η y which proves (5).(6) By [19, Corollary 3.4, Chapter V], it suffices to prove that ÿ i,j A y i E p a ˚ i a j q y i ξ | ξ E ě a , . . . , a n P A , all y , . . . , y n P M and every ξ P H . We have ÿ i,j A y i E p a ˚ i a j q y i ξ | ξ E “ ÿ i,j A E pr a i y i s ˚ r a j y j sq ξ | ξ E “ m ´ ÿ i,j ξ ˚ rp a i y i q ˚ p a j y j qs ˚ ξ ¯ ě ˜ÿ i,j ξ ˚ rp a i y i q ˚ p a j y j qs ˚ ξ ¸ p u q ě u P U M . Indeed, set C “ ¨˚˚˚˝ a y . . . a y . . . a n y n . . . ˛‹‹‹‚ P M n p B p H qq . Then C ˚ C “ ¨˚˚˚˝ ř i,j p a i y i q ˚ p a j y j q . . .
00 0 . . . . . . ˛‹‹‹‚ , and if ζ u “ ¨˚˚˚˝ u ˚ ξ ˛‹‹‹‚ , WEAKLY) ALMOST PERIODIC FUNCTIONS AND VON NEUMANN ALGEBRAS 9 we have ˜ÿ i,j ξ ˚ rp a i y i q ˚ p a j y j qs ˚ ξ ¸ p u q “ x C ˚ Cζ u | ζ u y ě u P U M . (cid:3) The following result describes completely the space wap H p B p H qq . Theorem 2.5.
We have the following equality: wap H p B p H qq “ K p H q ` C . In particular, if wap H p B p H qq “ B p H q then H is finite-dimensional.Proof. Separability of H implies that every selfadjoint operator is a sum D ` K where D is a selfadjoint, diagonal operator and K “ K ˚ P K p H q (itis a theorem due to Weyl; see [4, Proposition 4]). Hence it suffices to provethat if D “ D ˚ P wap H p B p H qq then D “ D c ` λ where D c is a selfadjoint,compact operator and λ P R . We assume that H is infinite-dimensional andwe set U p H q : “ U B p H q . The proof is divided into three steps.(i) Let D “ D ˚ P wap H p B p H qq be a selfadjoint, diagonal operator. Then D has at most one eigenvalue whose associated eigenspace is infinite-dimensio-nal. Indeed, assume on the contrary that D “ D ˚ has two eigenvalues λ “ λ whose associated eigenspaces H and H are infinite-dimensional.We claim that D does not belong to wap H p B p H qq . In order to prove that,we are going to apply Proposition 5.3. Thus, let p ε j q j ě (resp. p δ j q j ě ) bean orthonormal basis of H (resp. H ). For all integers i, j ě
1, we defineselfadjoint, unitary operators u i and v j as follows: their restrictions to theorthogonal complement p H ‘ H q K is the identity, and then set u i ε k “ ε k , k ď iδ k , k ą i and u i δ k “ δ k , k ď iε k , k ą i. Next, v j exchanges ε and ε j , and v j ξ “ ξ for all ξ P t ε , ε j u K . Then, forfixed i , we have u i v j ε “ u i ε j “ δ j for j ą i , hence ε ‹ D ‹ ε p v j u i q “ x v j u i Du i v j ε | ε y “ x Dδ j | δ j y “ λ for every j ą i , and thus lim j ε ‹ D ‹ ε p v j u i q “ λ for every i . This giveslim i p lim j ε ‹ D ‹ ε p v j u i qq “ λ . Next, let us fix j ě
1. If i ą j , one has u i v j ε “ u i ε j “ ε j , and we get lim i ε ‹ D ‹ ε p v j u i q “ λ . This implies thatlim j p lim i ε ‹ D ‹ ε p v j u i qq “ lim i p lim j ε ‹ D ‹ ε p v j u i qq , and Proposition 5.3 implies that ε ‹ D ‹ ε R WAP p U p H qq , which provesthe claim. In particular, if D has finite spectrum, then it has exactly oneeigenvalue λ whose associated eigenspace is infinite-dimensional. This im-plies that D ´ λ is a finite-rank operator.(ii) Assume next that D admits infinitely many distinct eigenvalues p λ k q k ě such that at most one of them, say λ , has an infinite-dimensional eigenspace.If it is the case, replacing D by D ´ λ , we assume that all eigenspaces of allnon-zero eigenvalues are finite-dimensional. As the spectrum of D is infiniteand bounded, it possesses at least one accumulation point. In fact, we claimthat it possesses exactly one such point. Indeed, if p λ k q has two distinctaccumulation points α “ β , then there are sequences p u i q , p v j q Ă U p H q anda vector ξ P H such thatlim i p lim j ξ ‹ D ‹ ξ p v j u i qq “ α “ β “ lim j p lim i ξ ‹ D ‹ ξ p v j u i qq , and D does not belong to wap H p B p H qq . Indeed, let p α k q k ě (resp. p β k q k ě )be a sequence of distincts eigenvalues of D which converges to α (resp. β ), and such that α k “ β ℓ for all k, ℓ . Choose for every k a norm-oneeigenvector ε k (resp. δ k ) of α k (resp. β k ), so that the sequences p ε k q and p δ k q are orthonormal. Then, for all i, j ě
1, define u i and v j exactly as inPart (i). Then for every i , one has ε ‹ D ‹ ε p v j u i q “ β j Ñ β as j Ñ 8 , sothat lim i p lim j ε ‹ D ‹ ε p v j u i qq “ β and similarly, lim j p lim i ε ‹ D ‹ ε p v j u i qq “ α, which proves the claim.(iii) Finally, we are left with the case where the set of distinct non-zeroeigenvalues of D is infinite and has a unique accumulation point λ , and suchthat every eigenspace is finite-dimensional (except maybe the one associatedto λ ). Thus D ´ λ has the same properties as D , but as 0 is the onlyaccumulation point of D ´ λ and as all non-zero eigenvalues have finite-dimensional eigenspaces, this proves that D ´ λ is a compact operator. (cid:3) Proposition 2.6.
Assume that H is infinite-dimensional, and let M Ă B p H q be either a diffuse von Neumann algebra, or M “ B p H q , and let E :wap H p M q Ñ M be the positive map of Theorem 2.4. Then K p H q Ă ker E .Proof. Since E is a continuous, positive map, it suffices to prove that, for all ζ, ξ P H , x E p T ζ,ζ q ξ | ξ y “ T ζ,ζ is therank one operator defined by T ζ,ζ p ξ q “ x ξ | ζ y ζ for every ξ P H . WEAKLY) ALMOST PERIODIC FUNCTIONS AND VON NEUMANN ALGEBRAS 11
Thus, fix ζ, ξ P H ; we have for every u P U M : ξ ‹ T ζ,ζ ‹ ξ p u q “ x uT ζ,ζ u ˚ ξ | ξ y “ xx u ˚ ξ | ζ y uζ | ξ y“ x ξ | uζ yx uζ | ξ y “ |x uζ | ξ y| . The function ϕ on U M defined by ϕ p u q : “ x uζ | ξ y for every u P U M belongsto WAP p U M q by the proof of property (c) of Theorem 2.4. As |x uζ | ξ y| ď } ζ }} ξ }|x uζ | ξ y| for every u P U M , it suffices to prove that m p| ϕ |q “
0. Its proof is inspiredby that of [2, Theorem 1.3]. Let ε ą
0. By condition (d) of Theorem 5.5,there exist v , . . . , v m P U M and t , . . . , t m ą ř j t j “ ˇˇˇ ÿ j t j | ϕ p v ˚ j u q| ´ m p| ϕ |q ˇˇˇ ă ε { u P U M . Assume first that M is diffuse; then there exists asequence p u n q Ă U M such that u n Ñ n suchthat | ϕ p v ˚ j u n q| ă ε { j , so that0 ď ÿ j t j | ϕ p v ˚ j u n q| ă ε { . This implies that0 ď m p| ϕ |q ď ˇˇˇ ÿ j t j | ϕ p v ˚ j u n q| ´ m p| ϕ |q ˇˇˇ ` ÿ j t j | ϕ p v ˚ j u n q|ď ε. Finally, assume that M “ B p H q , and let F Ă H be the linear span of the set t v ξ, . . . , v m ξ u , which is finite-dimensional. Since H is infinite-dimensional,there exists η P F K such that } η } “ } ζ } . Hence, there exists a unitaryoperator u on H such that uζ “ η K v j ξ for every j “ , . . . , m . Thisimplies that ϕ p v ˚ j u q “ j and thus that m p| ϕ |q ă ε { (cid:3) We end the present section with three remarks. The last one is inspiredby [5, Remarques (i)].
Remark . Let µ P C b,r p U M q ˚ be a continuous linear functional on the C ˚ -algebra C b,r p U M q . Then, the effect of µ on B p H q can be described asfollows: as in the proof of Theorem 2.4, there exists a unique bounded, linearmap E µ : B p H q Ñ B p H q such that x E µ p T q ξ | η y “ µ p ξ ‹ T ‹ η q for all T P B p H q and ξ, η P H . Equation (2.4) implies that E µ is an M -bimodular map, i.e. E µ p x T y q “ x E µ p T q y for all T P B p H q and x , y P M . Remark . If M is such that wap H p M q “ B p H q , then M is ApproximatelyFinite-Dimensional. Indeed, if it is the case, as in [5], the map E is a condi-tional expectation from B p H q onto M , and M has Schwartz’s property P. Moreover, as will be proved in § M is finite since it is then contained inwap H p M q : see Theorem 3.1 and Remark 3.4. Remark . Let H be an infinite-dimensional Hilbert space and let A Ă B p H q be a unital C ˚ -algebra which has no tracial states. Let M Ă B p H q be a von Neumann algebra containing A . Then A Ć wap H p M q . Indeed,otherwise, we would have E p uau ˚ q “ E p a q for all a P A and u P U M . Thiswould imply that E p xy ´ yx q “ x, y P A , but this leads to acontradiction since 1 is a finite sum of commutators of elements of A by [15,Theorem 1].3. The case of finite von Neumann algebras
In this section, we consider a von Neumann algebra M Ă B p H q and wedenote by M ˚ its predual. We denote by Int p M q the group of its innerautomorphisms and, for v P U M , we denote by Ad p v q P Int p M q the auto-morphism given by Ad p v qp x q “ vxv ˚ for every x P M .Let B p M q (resp. B ˚ p M q ) denote the Banach space of all bounded (resp.ultraweakly continuous) linear operators on M . The weak ˚ topology on B p M q is the σ p B p M q , M b γ M ˚ q -topology, where M b γ M ˚ is the projectivetensor product of M and M ˚ (see [19, Chapter IV]). In fact, if p Φ i q Ă B p M q is a bounded net, then it converges weakly ˚ to Φ P B p M q if and only if ϕ p Φ i p x qq “ x Φ i , x b ϕ y Ñ ϕ p Φ p x qq “ x Φ , x b ϕ y for all x P M and ϕ P M ˚ .If M is finite, we denote by Ctr M its canonical centre-valued trace. Inthis case, it follows from (the proof of) [19, Theorem V.2.4] that the weak ˚ closure of Int p M q in B p M q is contained in B ˚ p M q . Thus, as Int p M q isbounded, it is relatively weakly ˚ compact in B ˚ p M q (see [19, Exercise 6, p.333].)This allows us to prove the following theorem. Theorem 3.1.
The von Neumann algebra M is finite if and only if M Ă wap H p M q . In other words, M is finite if and only if, for every x P M ,all its coefficient functions ξ ‹ x ‹ η are weakly almost periodic. If it is thecase, wap H p M q contains C ˚ p M, M q , the C ˚ -algebra generated by M and M in B p H q , and the restriction E M of E to C ˚ p M, M q has the additionalproperties: (i) The restriction of E M to M coincides with the canonical centre-valued trace Ctr M . (ii) E M : C ˚ p M, M q Ñ M is a conditional expectation.Proof. If M Ă wap H p M q , then the restriction of E to C ˚ p M, M q satisfiescondition (5) in Theorem 2.4, and this implies that E p xy q “ E p yx q for all x, y P M . Furthermore, if x P M and u P U M , one has by property(3) in Theorem 2.4: u E p x q u “ E p u xu q “ E p x q , which means that the WEAKLY) ALMOST PERIODIC FUNCTIONS AND VON NEUMANN ALGEBRAS 13 restriction of E to M maps M onto its centre Z p M q . Then [7, Corollary 3,Part III, Chapter 8] shows that M is a finite von Neumann algebra.Conversely, suppose that M is finite. Let x P M and ξ, η P H . We mustprove that the coefficient function ξ ‹ x ‹ η belongs to WAP p U M q . It followsfrom [12, Proposition 7] or by Proposition 5.4 of the appendix that it sufficesto prove that the right orbit p ξ ‹ x ‹ η q U M is relatively weakly compact in C b,r p U M q . But we have, by equation (2.3): p ξ ‹ x ‹ η q U M “ t ξ ‹ vxv ˚ ‹ η : v P U M u “ t ξ ‹ θ p x q ‹ η : θ P Int p M qu . Thus, from Grothendieck’s Theorem 5.2 in the appendix, it suffices to provethat if p u i q Ă U M and p θ j q Ă Int p M q are sequences such that the followingdouble limits ℓ : “ lim i p lim j p ξ ‹ θ j p x q ‹ η qp u i qq and ℓ : “ lim j p lim i p ξ ‹ θ j p x q ‹ η qp u i qq exist, then they are equal. Set ψ i “ Ad p u i q for every i . Then p ξ ‹ θ j p x q ‹ η qp u i q “ x u i θ j p x q u ˚ i ξ | η y “ x ψ i p θ j p x qq ξ | η y for all i, j . Extracting subse-quences if necessary, we assume that p θ j q converges weakly ˚ to some limit θ P Int p M q Ă B ˚ p M q , and that p ψ i q converges weakly ˚ to some limit ψ P Int p M q Ă B ˚ p M q . Denoting by ω ξ,η P M ˚ the normal linear func-tional y ÞÑ x yξ | η y , one has for every i on the one hand, because ψ i is anormal map,lim j x ψ i p θ j p x qq ξ | η y “ x ψ i p θ p x qq ξ | η y “ x ψ i ˝ θ, x b ω ξ,η y “ x ψ i , θ p x q b ω ξ,η y . Hence ℓ “ lim i x ψ i , θ p x q b ω ξ,η y “ x ψ ˝ θ p x q ξ | η y .On the other hand, one has for every j lim i x ψ i p θ j p x qq ξ | η y “ x ψ p θ j p x qq ξ | η y . As ψ is normal, we get ℓ “ lim j x ψ p θ j p x qq ξ | η y “ x ψ ˝ θ p x q ξ | η y “ ℓ . Hence M Ă wap H p M q .From now on, we assume that M is finite.(i) For every x P M , the map E is constant on the convex hull of t uxu ˚ : u P U M u , hence, by its boudedness, it is constant on the norm closure K x of thelatter. By [7, Theorem 1, Part III, Chapter 5], E p Ctr M p x qq “ E p x q , and asCtr M p x q P M X M , we have E p x q “ Ctr M p x q . This proves (i).(ii) follows from property (3) of Theorem 2.4. (cid:3) Remark . The relative compactness of Int p M q in B ˚ p M q is a special caseof the notion of G -finite von Neumann is defined in [13]; see also [18] and[22].The following example shows that the hypothesis that M is diffuse inProposition 2.6 cannot be removed. Even though it is a special case ofTheorem 3.1, we think it is worth being discussed. Example . Set H “ ℓ p N q , let p δ k q k P N be the natural orthonormal basisof H and let M “ A “ ℓ p N q be the atomic maximal abelian ˚ -subalgebraof B p H q acting by pointwise multiplication on H so that aδ k “ a p k q δ k forall a P A and k P N . We claim that wap H p A q “ B p H q and that E p T q P A isthe function k ÞÑ x
T δ k | δ k y for every T P B p H q .Indeed, let T P B p H q . In order to prove that it belongs to wap H p A q , itsuffices to verify that δ k ‹ T ‹ δ ℓ is weakly almost periodic for all k, ℓ P N .Thus, let us fix integers k and ℓ . As U A “ T N , where T denotes the unitcircle, we have for all u P U A : δ k ‹ T ‹ δ ℓ p u q “ x T u p k q δ k | u p ℓ q δ ℓ y “ x T δ k | δ ℓ y u p k q u p ℓ q . But u p k q u p ℓ q “ x δ k | uδ k yx uδ ℓ | δ ℓ y “ xx δ k | uδ k y uδ ℓ | δ ℓ y“ x u x u ˚ δ k | δ k y δ ℓ | δ ℓ y “ x uT δ ℓ ,δ k u ˚ δ k | δ ℓ y“ δ k ‹ T δ ℓ ,δ k ‹ δ ℓ p u q where we use the same notation as in the proof of Theorem 2.4(c) for rankone operators. Hence T P wap H p A q and x E p T q δ k | δ ℓ y “ x T δ k | δ ℓ y ¨ m p δ k ‹ T δ ℓ ,δ k ‹ δ ℓ q . Set ϕ k,ℓ “ δ k ‹ T δ ℓ ,δ k ‹ δ ℓ for short so that ϕ k,ℓ p u q “ u p k q u p ℓ q for every u P U A .If k “ ℓ , then ϕ k,k p u q “ u and m p ϕ k,k q “
1. If k “ ℓ , then wecan view ϕ k,ℓ as the continuous function on T defined by ϕ k,ℓ p z, w q “ zw .As T is a compact group, one has C p T q “ WAP p T q and the invariantmean on the latter coincides with the Haar measure. Hence, by property(d) of Theorem 5.5, we have m p ϕ k,ℓ q “ ij T zwdzdw “ . Remark . Assume that M is a finite von Neumann subalgebra of B p H q ,and let τ be a normal, faithful normalized trace on M . Set } x } : “ τ p x ˚ x q { for every x P M . Then the topology on U M is induced by the complete,bi-invariant metric p u, v q ÞÑ } u ´ v } . This means that every bounded,continuous function f on U M is uniformly right continuous if and only if itis uniformly left continuous, namely that the map v ÞÑ v ¨ f is continuous ifand only if the map v ÞÑ f ¨ v is. Suppose moreover that M is injective, andlet p M n q n ě be an increasing sequence of finite-dimensional C ˚ -subalgebrasof M , such that 1 P M n for every n and that their union M : “ Ť n M n isstrongly dense in M . Then, as proved in [5], C b,r p U M q has a bi-invariantmean µ , hence so does C b,r p U M q by the bilateral uniform continuity of allelements of the latter algebra. Consequently, even if we do not know whetherwap H p M q is equal to B p H q , the existence of µ implies the existence of apositive map E µ : B p H q Ñ M which has properties (1)–(6) of Theorem 2.4 WEAKLY) ALMOST PERIODIC FUNCTIONS AND VON NEUMANN ALGEBRAS 15 as well as those in Theorem 3.1. In particular, the restriction of E µ to M isequal to Ctr M .4. The case of almost periodic operators relative to M Let M Ă B p H q be a von Neumann algebra. Recall that ap H p M q is the setof all operators T P B p H q such that ξ ‹ T ‹ η P AP p U M q for all ξ, η P H . As inthe proof of Theorem 2.4, ap H p M q is an operator system which contains M .We remind the reader of characterizations of almost periodic functions ontopological groups in Theorem 5.6, which is taken from [6, Theorem 16.2.1].We are grateful to P. de la Harpe for having indicated that reference.We also need to recall definitions of diffuse and atomic von Neumannalgebras. Our reference on these notions is [17]. Denote by P p M q the set ofall orthogonal projections of M . An element e P P p M q is an atom in M if itsatisfies the equality: eM e “ C e . If it is the case, then its central cover z p e q (i.e. the smallest projection z of the centre Z p M q of M such that ze “ e ) isan atom of the centre Z p M q , thus M z p e q is a factor. Definition 4.1.
The von Neumann algebra M is called atomic if, for everynon-zero projection f P P p M q , there exists an atom e P P p M q such that e ď f . If M contains no atoms, then it is called diffuse .As a consequence of the above facts, if M is atomic and finite, then it isa direct sum of finite-dimensional factors. Lemma 4.2.
Let M be a von Neumann algebra. Then there is a uniquecentral projection z such that M z is diffuse and M p ´ z q is atomic.Proof. Uniqueness of z is straightforward to check.Concerning the existence of z , if M is diffuse, we set z “
1. Thus, let usassume that M has atoms. We define then Z which is the set of familiesof atoms p e i q i P I Ă M such that z p e i q z p e j q “ for all i “ j . The set Z isnon-empty since M has atoms, and is ordered by inclusion: p e i q i P I ď p f j q j P J if and only if I Ă J and e i “ f i for every i P I . By Zorn’s Lemma, we choosea maximal element p e i q i P I P Z and set1 ´ z “ ÿ i P I z p e i q . Then one verifies easily that
M z is diffuse and that M p ´ z q is atomic, bymaximality. (cid:3) We need the following definition, which is due to J. von Neumann [20, p.482]; see also [21].
Definition 4.3.
A topological group G is minimally almost periodic if itsonly continuous, finite-dimensional, irreducible, unitary representation is theone-dimensional trivial representation. It follows from [6, Theorem 16.2.1] that G is minimally almost periodic ifand only if AP p G q “ C . For the reader’s convenience, we recall some partsof the latter theorem in Theorem 5.6.The following result, which is used below for finite von Neumann algebras,seems to be new, as far as we know. We are grateful to P. de la Harpe forhis suggestions in the treatment of the general case. In order to state it,let us recall from [19, Chapter V] that an arbitrary von Neumann algebra M acting on the separable, infinite-dimensional Hilbert space H admits thefollowing direct sum decomposition: M “ M I f ‘ M I ‘ M II ‘ M III where(1) M I f is a direct sum M I f “ à j ě A j b M j p C q with A j abelian and, for every j ě
1, where M j p C q is the finite factorof type I j ;(2) M I “ A b B p K q with A abelian and where K is the separable,infinite-dimensional Hilbert space and where N b P denotes the usualvon Neumann algebra tensor product of the von Neumann algebras N and P ;(3) M II and M III are the type II and III components of M respectively.They are both diffuse.Lemma 4.2 implies that each A j in (1) is a direct sum A j “ C j ‘ D j where C j is atomic, hence isomorphic either to ℓ p N q or to C m j for somepositive integer m j , and where D j is diffuse. Rearranging the componentsof M I f , we see that it is expressed as follows:(4.1) M I f “ à k ě M k where M is a direct sum of tensor products M “ À ℓ ě D ℓ b M p ℓ p C q with D ℓ abelian and diffuse for every ℓ , and where M k “ M n k p C q with 1 ď n k ă 8 for every k ě Theorem 4.4.
Let M Ă B p H q be a von Neumann algebra acting on theseparable, infinite-dimensional Hilbert space H . Let M “ M I f ‘ M I ‘ M II ‘ M III be the above decomposition of M , with M I f “ À k ě M k as in (4.1). Then theunitary group U M is minimally almost periodic, in other words AP p U M q “ C , if and only if the atomic part À k ě M k of M I f is equal to .Proof. If 1 “ ř ℓ z ℓ is a partition of the unity with z ℓ P P p Z p M qq for every ℓ , then U M decomposes as U M “ ź ℓ U Mz ℓ , WEAKLY) ALMOST PERIODIC FUNCTIONS AND VON NEUMANN ALGEBRAS 17 and U M is minimally almost periodic if and only if each U Mz ℓ is. Hence, ifthe atomic part À k ě M k of M I f is non-trivial, U M is not minimally almostperiodic.Thus the proof will be complete if we prove that all groups U M X for X P t , I , II , III u are minimally almost periodic.We divide the proof into three parts.(i) Assume that A is an abelian and diffuse von Neumann algebra, so thatit is ˚ -isomorphic to L r , s . It suffices to prove that the only continuouscharacter χ : U A Ñ T is the trivial one. By spectral theory, the subset S : “ ! exp ´ i kπm p ¯ : m ě , ď k ď m ´ , p P P p A q ) generates a dense subgroup of U A in the norm topology. Thus, let us fixa continuous character χ of U A , and let u “ exp p i kπp { m q P S . Choose ε ą m -th root ω of the unity such that | ω ´ | ă ε , and next choose δ ą | χ p v q´ | ă ε for every v P U A such that } v ´ } ă δ .Since A is diffuse, there exist pairwise orthogonal projections q , . . . , q n P P p A q such that p “ ř j q j and that ››› exp ´ i kπm q j ¯ ´ ››› ă δ for every j . Then χ ´ exp ´ i kπm q j ¯¯ is an m -th root of the unity which is atdistance within ε to 1, hence it is equal to 1, and finally χ ´ exp ´ i kπm p ¯¯ “ n ź j “ χ ´ exp ´ i kπm q j ¯¯ “ . (ii) If N is a diffuse von Neumann algebra, every unitary u P U N belongsto some maximal abelian ˚ -subalgebra A “ A p u q of N , which is necessarilydiffuse by maximality. Hence, if π is a continuous, finite-dimensional, irre-ducible, unitary representation of U N , and if u P U N , the restriction of π to U A p u q is a direct sum of irreducible ones, hence it is trivial by Part (i).In particular, π p u q “
1. As this is true for every u , the representation π is trivial. As the components M II and M III of M are diffuse, this provesthat the groups U M X are minimally almost periodic for X P t , II , III u . Thisalso proves that U N is minimally almost periodic if N “ D b B p K q with D abelian and diffuse.(iii) Assume at last that N is a von Neumann algebra of type I with atomiccentre. By [19, Theorem V.1.27], N “ A b B p K q with A abelian and atomic.It suffices to prove that the unitary group U p K q is minimally almost periodic,which seems to be known by the experts. We sketch a proof for the sake ofcompleteness. Let us first recall the description of all strongly continuous,irreducible unitary representations of U p K q (see for instance [1, Proposition9]): Let ρ : U p K q Ñ U p K q be the tautological representation of U p K q . Thenevery strongly continuous, irreducible unitary representation π of U p K q is unitarily equivalent to a subrepresentation of the representation ρ b k b ¯ ρ b ℓ of U p K q on K b k b ¯ K b ℓ for some integers k, ℓ ě
0, where the case k ` ℓ “ p π, L q is a non-trivial strongly continuous, irreducible unitary represen-tation of U p K q , then, conjugating π by a suitable unitary, we assume that L contains a unit vector of the form ξ b k b ¯ ξ b ℓ with k ` ℓ ą
0. Then theorbit π p U p K qq ξ b k b ¯ ξ b ℓ “ tp uξ q b k b p uξ q b ℓ : u P U p K qu contains an infiniteorthonormal system, and π is infinite-dimensional. (cid:3) We are ready to determine the von Neumann algebras M Ă B p H q suchthat M Ă ap H p M q . Theorem 4.5.
Let M Ă B p H q be a von Neumann algebra. Then M is con-tained in ap H p M q if and only if M is isomorphic to a direct sum À k ě M k where M is an abelian, diffuse von Neumann algebra, and where M k is afinite-dimensional factor for every k ě .Proof. Assume first that M is contained in ap H p M q . As the latter spaceis contained in wap H p M q , M is a finite von Neumann algebra by Theorem3.1. Let z be the central projection such that M z is diffuse and M p ´ z q is atomic. We are going to prove that M z “ Z p M q z . In order to dothat, let x “ xz P M z . It suffices to prove that ξ ‹ x ‹ η is constantfor all ξ, η P H . Indeed, if it is the case, then x P M X M z “ Z p M q z which is abelian and diffuse. Thus, let us fix ξ, η P H . From the equality ξ ‹ x ‹ η “ ξ ‹ xz ‹ η “ p zξ q ‹ x ‹ p zη q , we assume further that ξ “ zξ and η “ zη . Replacing M by M z , we assume henceforth that M is diffuse.As x P ap H p M q , the orbit U M ξ ‹ x ‹ η is relatively compact. By Theorems4.4 and 5.6, ξ ‹ x ‹ η is constant.Conversely, if M is isomorphic to a direct sum M ‘ À k ě M k where M isan abelian, diffuse von Neumann algebra and each M k is a factor of type I n k ,say, with 1 ď n k ă 8 for every k , let p z k q k ě be the partition of 1 formedby the central projections such that M z k is isomorphic to M k for all k ě H “ À k ě H k with H k “ z k H for all k . Let then x “ À k ě x k P M and ξ “ À k ξ k , η “ À k η k P H . Then with respect to the norm topology in C b,r p U M q , ξ ‹ x ‹ η “ x x ξ | η y ` lim N Ñ8 ξ p N q ‹ x p N q ‹ η p N q with ξ p N q ‹ x p N q ‹ η p N q P ap p U M p N q q where M p N q “ À Nk “ M k is finite-dimensional. Indeed, one has for every u “ À k u k P U M : ξ ‹ x ‹ η p u q “ N ÿ k “ x u k xz k u ˚ k ξ k | η k y ` ÿ k ą N x u k xz k u ˚ k ξ k | η k y and ˇˇˇ ÿ k ą N x u k x k u ˚ k ξ k | η k y ˇˇˇ ď } x } ÿ k ą N } ξ k }} η k } Ñ N Ñ8 ř k } ξ k } ă 8 and ř k } η k } ă 8 . (cid:3) WEAKLY) ALMOST PERIODIC FUNCTIONS AND VON NEUMANN ALGEBRAS 19
Contrary to wap H p M q which always contains at least K p H q ` C , ap H p M q can be very small, as the following proposition shows. Proposition 4.6.
Suppose that H is infinite-dimensional. Then ap H p B p H qq “ C . Proof.
We already know from Theorem 2.5 that ap H p B p H qq Ă K p H q ` C .Thus, let T P K p H q be a non-zero, positive operator. Let p λ j q j ě Ă R ˚` bethe sequence of positive eigenvalues of T so that λ ě λ ě . . . and λ j Ñ j Ñ8
0. Let p ε j q j ě Ă H be an orthonormal system such that T ε j “ λ j ε j for every j . Then, denoting by P the orthogonal projection onto ker p T q and by P ξ the rank-one projection onto C ξ for every unit vector ξ , we have T “ ÿ j ě λ j P ε j and T P “ P T “ u, v P U p H q : v ¨ ε ‹ T ‹ ε p u q “ p vε q ‹ T ‹ p vε qp u q “ x T u ˚ vε | u ˚ vε y“ ÿ j ě λ j x P ε j p u ˚ vε q| u ˚ vε y“ ÿ j ě λ j |x u ˚ vε | ε j y| “ ÿ j ě λ j |x vε | uε j y| . Let then N ą ď λ j ď λ { j ě N , anddefine p v n q n ě N Ă U p H q such that v n is the identity on P H and v n ε j “ $’&’% ε j j “ , nε n j “ ε j “ n. Then, as ε ‹ T ‹ ε p q “ λ , and as v ˚ n v m ε “ v ˚ n ε m “ ε m for all n, m ě N , n “ m , we get: } v m ¨ ε ‹ T ‹ ε ´ v n ¨ ε ‹ T ‹ ε } “ } ε ‹ T ‹ ε ´ v ˚ m v n ¨ ε ‹ T ‹ ε } ě | ε ‹ T ‹ ε p q ´ ε ‹ T ‹ ε p v ˚ n v m q|“ ˇˇˇ λ ´ ÿ j λ j x ε m | ε j y ˇˇˇ “ λ ´ λ m ě λ { ą . This shows that the orbit U p H q ε ‹ T ‹ ε is not relatively compact in thenorm topology of C b p U p H qq , hence that T R ap H p B p H qq . (cid:3) Remark . As for wap H p M q , we do not know whether ap H p M q is a C ˚ -algebra. Appendix: Weakly almost periodic functions on topologicalgroups
As promised in §
1, the aim of this appendix is to give a sketched proofof the existence of a unique invariant mean on WAP p G q , where G is anarbitrary topological group. We keep notation that were settled in § Theorem 5.1. [12, Th´eor`eme 5]
Let Ω be a compact space and let A Ă C p Ω q be a bounded set. Then A is relatively weakly compact if and only if, forevery sequence p f n q Ă A , there exists a subsequence p f n k q and an element h P C p Ω q such that lim k f n k p ω q “ h p ω q for every ω P Ω .Proof. (Sketch) The proof rests on Eberlein-Smulian theorem [9, TheoremA.12] which states that if A is a bounded subset of a Banach space X , then A is relatively weakly compact if and only if every sequence in A has asubsequence which converges weakly in X .Thus, if a bounded sequence p f n q converges pointwise to the limit h , then,by Lebesgue theorem, ş f n dµ Ñ ş hdµ for every regular complex measure µ on Ω. Compactness of Ω implies that every continuous linear functionalon C p Ω q is such a measure, hence relative compactness in the pointwiseconvergence topology implies relatively weak compactness. The converse isobvious, as every linear form of the type f ÞÑ f p ω q is weakly continuous. (cid:3) Using the Stone-Cech compactification of G , A. Grothendieck gets thefollowing theorem. Theorem 5.2. [12, Th´eor`eme 6]
Let G be an arbitrary topological group.Then a bounded subset A of C b p G q is weakly relatively compact if and onlyif there do not exist sequences p x i q Ă G and p f j q Ă A such that lim i p lim j f j p x i qq and lim j p lim i f j p x i qq both exist and are different. As a consequence, he gets the following criterion for weakly almost peri-odic functions on topological groups.
Proposition 5.3.
Let G be a topological group and let f P C b,r p G q . Then f is weakly almost periodic if and only if there do not exist sequences p x i q , p y j q in G such that lim i p lim j f p x i y j qq and lim j p lim i f p x i y j qq both exist and are different. In particular, the left orbit Gf is relativelyweakly compact if and only if the right orbit f G is. WEAKLY) ALMOST PERIODIC FUNCTIONS AND VON NEUMANN ALGEBRAS 21
Proposition 5.4.
Let G be as above. The set WAP p G q is a unital C ˚ -subalgebra of C b,r p G q which is left- and right-invariant under translations of G .Proof. It is clear that WAP p G q contains all constant functions, and that f P WAP p G q if f P WAP p G q . Moreover, for fixed g P G , the maps f ÞÑ g ¨ f and f ÞÑ f ¨ g are clearly weakly continuous, hence g ¨ f, f ¨ g P WAP p G q if f P WAP p G q .In order to prove that WAP p G q is a C ˚ -algebra, we are going to applyTheorem 5.1. In order to do that, let Ω be the Gelfand spectrum of theunital C ˚ -algebra C b,r p G q . It is a compact space, and Gelfand transform f ÞÑ ˆ f : χ ÞÑ χ p f q is a ˚ -isomorphism from C b,r p G q onto C p Ω q , and theweak topology on C b,r p G q corresponds to that on C p Ω q . Furthermore, G acts continuously on Ω by g ¨ χ p f q “ χ p g ´ ¨ f q . Indeed, if χ i Ñ χ and g j Ñ
1, one has | χ p f q ´ χ i p g ´ j ¨ f q| ď | χ p f q ´ χ i p f q| ` | χ i p f q ´ χ i p g ´ j ¨ f q|ď | χ p f q ´ χ i p f q| ` } f ´ g ´ j ¨ f } “| χ p f q ´ χ i p f q| ` } g j ¨ f ´ f } Ñ i, j Ñ 8 . Thus the image { WAP p G q of WAP p G q under the Gelfandtransform is exactly the set of elements f P C p Ω q for which Gf is relativelyweakly compact. The fact that WAP p G q is a ˚ -algebra is a straightforwardconsequence of Theorem 5.1.Let us prove finally that WAP p G q closed in C b,r p G q or, what amounts tobe the same, that { WAP p G q is closed in C p Ω q : let p f n q n ě Ă { WAP p G q be asequence which converges to f P C p Ω q . Let us show that Gf is relativelyweakly compact. In order to do that, let p g k q k ě Ă G be a sequence. Weare going to prove that there exists a subsequence p g k j q j ě Ă p g k q and anelement h P C p Ω q such that g k j ¨ f p ω q Ñ h p ω q for every ω P Ω. By thestandard diagonal process, there exist a subsequence p g k j q Ă p g k q and asequence p h ℓ q ℓ ě Ă C p Ω q such thatlim j Ñ8 g k j ¨ f ℓ p ω q “ h ℓ p ω q for every ω P Ω and every ℓ ě
1. It is easy to prove that p h ℓ q is a Cauchysequence in C p Ω q and then that p h ℓ q converges in norm to a limit denotedby h P C p Ω q . Finally, one proves that, for every ω P Ω, g k j ¨ f p ω q Ñ j Ñ8 h p ω q . (cid:3) Here is now the promised theorem on the existence of the unique bi-invariant mean on WAP p G q . Theorem 5.5.
There exists a unique linear functional m : WAP p G q Ñ C with the following properties: (a) m p f q ě for every f ě ; (b) m p q “ ; (c) m p g ¨ f q “ m p f ¨ g q “ m p f q for all f P WAP p G q and g P G ; (d) for every f P WAP p G q and every ε ą , there exists a convex com-bination ψ : “ ř mj “ t j g j ¨ f (with g j P G and t j ě , ř j t j “ )such that } ψ ´ m p f q} ă ε, and there exists a convex combination ϕ : “ ř i s i f ¨ h i (with h i P G and s i ě , ř i s i “ ) such that } ϕ ´ m p f q} ă ε. Proof.
For f P WAP p G q , let us denote by Q l p f q “ co p Gf q the norm closedconvex hull of Gf and similarly Q r p f q “ co p f G q . The group G acts byleft translations on Q l p f q which are affine transformations and are weaklycontinuous since, for every fixed g , the map f ÞÑ g ¨ f is linear and iso-metric. Moreover, if ψ , ψ P Q l p f q are such that ψ “ ψ , then 0 Rt g ¨ ψ ´ g ¨ ψ : g P G u }¨} and thus the action of G on Q l p f q is distal. ByRyll-Nardzewski Theorem, there exists c l p f q P Q l p f q such that g ¨ c l p f q “ c l p f q for every g P G . This means that c l p f q is constant.Similarly, G acts on the right on Q r p f q , and by the same arguments, Q r p f q contains a constant c r p f q . Using convex approximations of both constants,it is easy to check that Q l p f q and Q r p f q contain the same constant which isthen unique.Thus, by definition, m p f q “ c l p f q “ c r p f q for every f P WAP p G q .If a linear form m : WAP p G q Ñ C satisfies (a) and (b) and if m p g ¨ f q “ m p f q for all f P WAP p G q and g P G , then m is constant on Q l p f q andwe infer that m p f q “ c l p f q . If m is another left-invariant mean, one hasnecessarily m p f q “ m p f q by the above remarks. This proves uniqueness of m . Furthermore, the fact that m p f q belongs to Q l p f q X Q r p f q implies (d)and then the right invariance of m .Properties (a), (b), (c) and (d) are obvious, as well as the fact that m p αf q “ α m p f q for all α P C and f P WAP p G q . We are left to provethat, for f , f P WAP p G q , one has m p f ` f q “ m p f q ` m p f q . Let us fix ε ą
0. There exist s , . . . , s m ą ř i s i “
1, and g , . . . , g m P G such that ››› ÿ i s i g i ¨ f ´ m p f q ››› ď ε . for every g P G . But m p f q “ m ´ ř i s i g i ¨ f ¯ . Indeed, ř i s i g i ¨ f P Q l p f q ,and as the latter set is convex, we have Q l ´ ÿ i s i g i ¨ f ¯ Ă Q l p f q . Hence the constant in the left-hand convex set is equal to the one in Q l p f q .Thus there exist t , . . . , t n ą ř j t j “
1, and h , . . . , h n P G such WEAKLY) ALMOST PERIODIC FUNCTIONS AND VON NEUMANN ALGEBRAS 23 that ››› m p f q ´ ÿ i,j s i t j p h j g i q ¨ f ››› “ ››› m ´ ÿ i s i g i ¨ f ¯ ´ ÿ j t j h j ¨ ´ ÿ i s i g i ¨ f ¯››› ď ε . As ř i,j s i t j “
1, one has ÿ i,j s i t j p h j g i q ¨ r f ` f s P Q l p f ` f q and ››› ÿ i,j s i t j p h j g i q ¨ r f ` f s ´ m p f q ´ m p f q ››› ď ››› ÿ i,j s i t j p h j g i q ¨ f ´ m p f q ››› ` ››› ÿ i,j s i t j p h j g i q ¨ f ´ m p f q ››› ď ÿ j t j ››› ÿ i s i h j ¨ g i ¨ f ´ m p f q ››› ` ε “ ÿ j t j ››› h j ¨ ´ ÿ i s i g i ¨ f ´ m p f q ¯››› ` ε ď ε. This shows that m p f q` m p f q P Q l p f ` f q . The proof is now complete. (cid:3) We end the appendix with the following result on almost periodic func-tions on the topological group G which is used in Theorems 4.4 and 4.5. Itis part of [6, Theorem 16.2.1]. Theorem 5.6.
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